CC BY
17
UDC 621.3
https://doi.Org/10.20998/2074-272X.2022.1.07
O. Oualah, D. Kerdoun, A. Boumassata
Comparative study between sliding mode control and the vector control of a brushless doubly fed reluctance generator based on wind energy conversion systems
Introduction. Nowadays, global investment in renewable energy sources has been growing intensely. In particular, we mention here that wind source of energy has grown recently. Purpose. Comparative study between sliding mode control and vector control of a brushless doubly fed reluctance generator based on wind energy conversion systems. Methods. This paper deals with conceptual analysis and comparative study of two control techniques of a promising low-cost brushless doubly-fed reluctance generator for variable-speed wind turbine considering maximum power point tracking. This machine's growing interest because of the partially rated power electronics and the high reliability of the brushless design while offering performance competitive to its famous spring counterpart, the doubly-fed induction generator. We are particularly interested in comparing two kinds of control methods. We indicate here the direct vector control based on Proportional-Integral controller and sliding mode controller. Results. Simulation results show the optimized performances of the vector control strategy based on a sliding mode controller. We observe high performances in terms of response time and reference tracking without overshoots through the response characteristics. The decoupling, the stability, and the convergence towards the equilibrium are assured. References 29, figures 10. Key words: wind energy, brushless doubly fed reluctance generator, vector control, sliding mode controller, maximum power point tracking.
Вступ. HuHi глобальш твестицп у вiдновлюванi джерела енергп cmpiMKO зростають. Зокрема, звернемо увагу, що остантм часом мае мкце зростання втряних джерел енергп. Мета. Порiвняльне до^дження мiж управлтням ковзним режимом та векторним управлтням безщткового реактивного генератора з подвтним живленням на основi систем перетворення енергп втру. Методи. Стаття присвячена концептуальному аналiзу та порiвняльному до^дженню двох методiв управлтня перспективним недорогим безщтковим реактивним генератором з подвтним живленням для втряног турбти зi змтною швидюстю з урахуванням вiдстеження точки максимальног потужностi. 1нтерес до щег машини зростае частково завдяки силовт електронщ, а також високт надiйностi безщтковог конструкци, яка при цьому пропонуе характеристики, порiвнянi з його загальновiдомим пружинним аналогом, тдукцтним генератором з подвтним живленням. Автори особливо зацжавлет у порiвняннi двох видiв методiв управлтня. Автори вказують тут на пряме векторне управлтня на основi пропорцтно-ттегрального регулятора та регулятора ковзного режиму. Результата. Результати моделювання показують оптимiзованi характеристики векторног стратеги управлтня на основi контролера ковзного режиму. Спостер^аються висок показники з точки зору часу вiдгуку та вiдстеження еталонних значень без перевищення показниюв вiдгуку. Розв'язка, стабтьтсть та прагнення дорiвноваги гарантуються. Бiбл. 29, рис. 10. Ключовi слова: енерпя BiTpy, безщггковий реактивний генератор з подвшним живленням, векторне управлшня, регулятор ковзного режиму, ввдстеження точки максимально!" потужность
1. Introduction. The earth climate can be seriously influenced by increasing green house gas emissions from conventional energy sources (such as oil and coal) as well as the growing concern of depletion of these resources in the near future [1].Wind turbine is one way to generate electricity from renewable sources, mainly because it is clean and economically viable [2]. At the same time, there has been a rapid development of related wind turbine [3].
The wind turbine is a very effective component in wind energy conversion systems (WECS) that converts the wind kinetic energy into mechanical energy that can be used to derive an electrical generator. The wind turbine generator converts the output mechanical energy of the wind turbine into electric power [4]. It can be connected either to standalone loads or connected to the utility grid [5].
Wind turbine generators used in wind energy conversion systems can be classified into two types: fixed speed wind turbine (FSWT) and variable speed wind turbine (VSWT). For the first one, the generator is connected directly to the grid without any intermediate of power electronic converters (PECs), and for the other one, the generator is connected through PECs [6-8]. Their findings show that the VSWT provides better energy capture over the FSWT because it can quickly adapt to a wide range of wind speed variations [9]. For this reason, the model considered in this research is the VSWT coupled to a Brushless Doubly Fed Reluctance Generator (BDFRG).
The brushless doubly-fed generator (BDFG) is a prominent economical solution to reliability and maintenance problems of brushes and slip-rings with the traditional doubly-fed induction generator (DFIG) in WECS [10, 11]. The BDFRG merits have inspired the research on its design, control, and grid integration aspects targeting WECS as one of the leading applications [12]. The strengths of this design include simple infrastructure, economical, reliable high efficiency, and robust power factor control capability [13]. Due to its high reliability these machines are also adaptable in air craft industry but some of design challenges include harsh aerodynamic and complex rotor design [14]. The background and fundamental structure of the Brushless Doubly Fed Reluctance Motor (BDFRM) was described in [15].
Control research [16], similar in context to that conducted on the BDFIG [17], and DFIG [18], has been carried out on the BDFRM(G) involving: scalar control, voltage vector-oriented control (VC) [19], direct torque and flux control (DTC) [20], torque, and reactive power control (TQC) [21, 22], direct power control (DPC) [23], sliding-mode power control [24], and even nonlinear multiple-input-multiple-output control [3]. Recently, sliding mode control (SMC) was integrated largely in the command of nonlinear systems [25].
The variable structure control possesses this robustness using the sliding mode controller. It offers
© O. Oualah, D. Kerdoun, A. Boumassata
excellent performances compared to unmodelled dynamics [26], insensitivity to parameter variation, external disturbance rejection, and fast dynamic [27].
The proposed system comprises a wind turbine, a BDFRG, an inverter, and a vector control based on PI controller and sliding mode controller.
This paper is organized as follows. A description of the studied WECS is presented in section 2. In section 3 the modeling of the wind turbine and the control of the maximum power point tracking (MPPT) are provided. Then, the BDFRG is modeled in Section 4. The vector control of BDFRG is given in Section 5. The sliding mode control is described in Section 6. The sliding mode control of the BDFRG is given in Section 7. Finally, simulation results and interpretations are presented in Section 8.
2. Description of the studied WECS. The WECS adopted here is shown in Fig. 1. The proposed system is constituted of a wind turbine, BDFRG, inverter, and power control. The BDFRG has two stator windings of different pole numbers. Generally, different applied frequencies (Fig. 1): the primary (power) winding is grid-connected, and the secondary (control) winding is converter fed. The performance of the proposed system has been tested to prove the MPPT control. The independent control of grid active and reactive powers using stator flux oriented control technique is used to test the ability to operate in two quadrant modes (sub-synchronous and super-synchronous modes).
Fig. 1. Conceptual diagram of the BDFRG-based WECS
3. Modeling of the wind turbine. The mechanical power extracted by the turbine from the wind is defined as:
1
P - 2-p-A ■ Cp •
(1)
where p is the air density; A = n-R is the rotor swept area; R is the turbine radius; Cp is the power coefficient; V is the wind speed.
The power coefficient Cp represents the aerodynamic efficiency of the wind turbine. It depends on the tip speed ratio X and the pitch angle fi (Fig. 2, 3). The tip speed ratio is given as:
¿ = ^, (2) V 9
where is the turbine speed.
For our example, the power coefficient Cp is given by the following equations
- N 21
^^ + 0.0068-^, (3)
c (à,ß)- 0.5176-
À - 0.4 ß-5
V Àx
■ e
where
_L - 1 - 0.035 À ~ À +0.08ß ß3 +1'
(4)
Fig. 2. Characteristics of the wind turbine power coefficient Cp with the tip speed ratio À at different values of the blade pitch angle ß
fit, rpm
Fig. 3. Power speed characteristics for different wind speeds
The maximum value of Cp (Cp max = 0.48) is for P = 0 degree and for X = 8.1. The turbine torque Tt can be written as:
Tt = Pt/fit. (5)
The mechanical speed of the generator fim and the torque of the turbine referred to the generator Tm are given by: torn =fit • G; \Tm = Tt/G, where G is the gearbox ratio.
The mechanical equation of the system can be characterized by:
(6)
J ¿"m dt
- Te - f "fi,
(7)
where J is the equivalent total inertia of the generator shaft; f is the equivalent total friction coefficient; Te is the electromagnetic torque.
To extract the maximum power from the wind turbine the electromagnetic torque command of the BDFRG Te.ref should be determined at the optimal value of the tip speed ratio and the corresponding maximum value of wind turbine power coefficient.
4. Mathematical model of BDFRG. The electrical equations of the BDFRG in the (d-q) Park reference frame are given by:
Upd - Rp - Ipd +
dO
p '1pd '
pd
dt dO
Upq - Rp ' Ipq + dt
-0
]p 'Vpq ;
pq
Usd - Rs - Isd +
dO
s ' 1sd '
+0-Vpd ; ^-(0r-0
(8)
■V,
sq
dt
Usq - Rs ' I sq + (wr -00-Vsd ,
sq
s sq
dt
¥pd - Lp ■ 1pd + Lm ■1 sd; V pq - Lp -1 pq - Lm '1sq ; ¥sd - Ls - 1sd + Lm - Jpd; Vsq - Ls - 1sq - Lm -1 pq •
(9)
The electromagnetic torque is expressed as:
3 * Pr * Lm ( j
Wpd * Isq + Wpq '
T = -
1 e
2 • Lr
■ *Isd )•
(10)
The active and reactive powers equations at the primary stator, the secondary stator, and the grid are written as, respectively:
"' ((pd '1 pd + Upq '1 pq ]
PP =
Qp 3 (u'Pq 1 pd U pd '1 pq )
P. = 3 (
• UA + U
' sd ' 1sd
sq
* Isq)
Qs = 2
sq
• Isd — Usd •1 sq );
IP = P + P ■
I g p s '
(11)
(12)
(13)
|Upd = 0;
(14)
[Upq = Up =®p Wp.
The expressions of the primary stator currents are
written as:
1 pd =
Ipq
Wpd
— Lm • I
'm '1 sd
Lm *1 sq
L„
(15)
By replacing these currents in the secondary stator fluxes equations, we obtain:
Wsd = &• Ls • Isd + Lm •Wpd;
lp
W sq = c* Ls •1 sq •
where a is the leakage coefficient defined by:
c = 1 —
L2
Lp * Ls
(16)
(17)
The secondary stator voltages can be written according to the secondary stator currents as:
Usd = Rs • Isd + Ls *c* ^ + eq ;
dis
(18)
sq
Usq = Rs 'Isq + Ls 'c" dt
+ ed + e,
<p>
where
eq = —Ds *c" Ls * Isq; ed = Ds *c" Ls *1 sd; Lm
ev = ®s ■ LT •Wp •
lp
(19)
The active and reactive stator powers of the BDFRG are expressed by:
3 U va ' Ln 2
(20)
[Qg = Qp + Qs;
where Up and Us are power and control windings voltages in the dq axis, respectively; Ip and Is are power and control windings currents in the dq axis, respectively; Rp and Rs are power and control windings resistances, respectively; Lp and Ls are leakage inductances of power and control windings, respectively; Lm is the mutual inductance between power and control windings; mp and ms are angular frequencies of the power and control windings, respectively; mr is the BDFRG mechanical rotor angular speed; wp and ws are power and control windings flux linkages in the dq axis, respectively.
5. Vector control of the BDFRG. In order to decouple the stator active and reactive powers, the primary stator flux vector will be aligned with d-axis ppd (ppd = pp and ppq = 0), and the stator voltages will be expressed by:
P =- pq * I
Qp =
3
L
U
sq '
p 2
pq
2 Dp*Lp
—3 • U 2
Lm I
pq ' L sd,
where s is the slip of the BDFRG.
The electromagnetic torque can be written as:
3 • P • L
" •Wpd • Isq . (21)
T =-
1 e
2 • Lr
Fig. 4. Block diagram of power control
For relatively weak sleep values and by neglecting the voltage drops, the grid active and reactive powers are simplified into:
p 3-(1 ~ s ]Upq • Lm
Pg 2• Lr ' sq;
Qg —•
3U
2
pq
2 (Dp • Lp
From (22) we have:
— 3 "(1 — s 2
2 • Lp
Isq 3 *(1 — s)Upq • Lm P
• U • Lm * i ,
^ pq j 1 sd-
Pg ;
(22)
r 3 *u 2
u pq 2 * Dp * Lp
\
—Qg
■2 * Lr
(23)
I = ±_
sd 3 *(1 — S)Upq * Lm
Substitute (23) in (18), we obtain:
Pg =-
U
sq
Rss
A * Ls *c Ls *c
* P —
rg
ed
A *Ls *c A *Ls *c
Q g
R
(24)
A * Ls *c Ls *c
■ Qg +
A *Ls*c A *Ls *c
L
e
<P
e
e
q
V
where
A -
2 ■ Lr
3-(1 - s )-Upq ■ L 3-U
e, - A - R s
pq ^m 2
pq
(25)
2-0p-Lp
In order to capture the optimal mechanical power, the control of the mechanical speed is applied (Fig. 5).
Control speed
Fig. 5. MPPT with the control of the speed
The reference value of the active power exchanged between the wind generator and the grid is generated by MPPT control, and it's given by:
Pg-ref Tem—ref' Q-mech. (26)
The reference grid reactive power Qgr-ref is fixed to zero value to maintain the power factor at unity. The detailed scheme of the studied system is illustrated in Fig. 6.
S -{x(()cr(x,t)- 0},
(28)
where
a(x,t) = [(x,t),a2(x,t),...,am(x,t) , (29)
where T denotes the transposed vector; S is called the sliding surface.
To bring the state variable to the sliding surfaces, the following two conditions have to be satisfied:
ct(x, t ) = 0; ct(x, t )= 0. (30)
The control law satisfies the precedent conditions is presented in the following form:
|U - Ueq + U„ ;
(31)
\U„ =-Kf • sgn(cr(x,t)),'
where U is the control vector; Ueq is the equivalent control vector; Un is the switching part of the control (the correction factor); Kf is the controller gain.
Ueq can be obtained by considering the condition for the sliding regimen, o(x, t) = 0. The equivalent control keeps the state variable on the sliding surface, once they reach it. For a defined function q>:
1, if p> 0; sgn(p)=jo, if <p = 0; (32)
-1, if p<0.
The controller described by (29) presents high robustness, insensitive to parameter fluctuations and disturbances. However, it will have high-frequency switching (chattering phenomena) near the sliding surface due to the (sgn) function involved. These drastic input changes can be avoided by introducing a boundary layer with width £. Thus replacing sgn(o(t)) by saturation function sat(o(t)/£) in (29) we have
U = Ueq - Kf ■ sat(cr(x,t)), (33)
where £ > 0
sai
ty-
isgn(^), if \<p\ > 1;
\cp, if \p< 1.
Consider a Lyapunov function:
V -1 -a2. 2
(34)
(35)
Fig. 6. The scheme of BDFRG using vector control
6. Sliding mode control. A Sliding Mode Controller (SMC) is a Variable Structure Controller (VSC) [28], which VSC includes several different continuous functions that can map plant state to a control surface, whereas switching among different functions is determined by plant state represented by a switching function [29]. The design of the control system will be demonstrated for the following nonlinear system:
X = f (x, t) + B(x, t )• u(x, t), (27)
where XeR" is the state vector; ueRm is the control vector; fx, t)eR"; B(x, t)eR"m.
From the system (16), it possible to define a set S of the state trajectories X such as:
From the Lyapunov theorem we know that if V is negative definite, the system trajectory will be driven and attracted toward the sliding surface and remain sliding on it until the origin is reached asymptotically:
tv 1 da . 11
V ----- a - a < -n - a
2 dt 11
(36)
where n is the constant positive value.
In this work we use the sliding surface proposed by J.J. Slotine
a
(x, t)-
n—1
(37)
where
x = [x, x, x, xn-1] is the state vector;
xd = [xd, xd, xd,
d r
e = x - x = [e, e, e.
.d-U
x ] is the desired state vector; , e"-1] is the error vector; y is a positive coefficient; n is the system order [28].
e
In BDFRG control using sliding mode theory the surface is chosen as a function of the error between the reference input signal and the measured signals.
7. Sliding mode control of the BDFRG. The mathematical model of the grid active and reactive powers in (24) has two vector controls, so we define two switching surfaces, and we set n = 1. According to (37) the switching surfaces of the stator powers are given by: js(P)= e{pg)= Pg.ref - pg;
is(Qg)= eQg )= Qg-ref - Qg. ( )
The second step consists of giving the structure of the vector control. One of the possible solutions is given by:
(39)
(40)
Uc = Ueq + Un ; I Usq - Usqeq + Usqn;
[Usd - Usdeq + U
sdn,
where Ueq (Usqeq, Us^eq) and Un (Usqn, Usdn) indicate the equivalent and discontinuous components of the control input vector Uc (Usq and Usd), respectively.
Ueq is calculated from S(x) - 0
Is (Pg)-Pg-ref - Pa - 0;
S (x)-0 % g f g (41)
[S(Qg)-Qg-ref -Qg -Substituting (24) in (41) we will have:
Pg -ref Qg-ref
Uy
sq
Rs
ed
VA*Ls a Ls a
A• Ls •a A• Ls •a
- 0;
(42)
-Us
sd
Rs
V
A • Ls •a Ls •a
Qg +
A • Ls • a A • Ls • a
- 0.
Pg-ref
Replaced Usq and Usd from (40) in (42) we obtain:
(sqeq + Usqn) R p - ed
L •a'Pg
e,
P
A• Lv •a
Qg-ref
(sdeq +Usdn) Rs q eq Qg +
A^Ls •a A^Ls •aj
- 0;
(43)
A• Lv •a
Ls •a
A• Ls •a A^Ls •aj
- 0.
In the permanent regime Un = 0 (Usqn = 0 and Usdn = 0). Replaced them in (43) and we extract Usqeq and Usdeq:
\Usqeq - A • Ls •a• Pg-ref + A • Rs • Pg-(ed - e()
(44)
vs * g ^p
\Usdeq --A • Ls Q g-ref - A • Rs • Qg +(eq + eu)
Usqn and Usdn are achieved by the condition: S (x )• s(x) < 0 ^ S (Pg ) sS (Pg )< 0 and S Qg ) S Qg )< 0 (45)
-1
A • a-Ls
1
A • a-Ls
■U.
sqn
•U,
sdn
•S )<0; •S (Qg )<0.
(46)
(47)
(48)
A simple used form of Un is a relay function: Un = -ksgn(S(x)); Usqn --ki • sgn(S(Pg)), [Usdn - -k2 • sgn(((Qg |
where k (k1, k2) must be positive to satisfy the previous condition.
Substituting (44) and (48) into (40) we obtain the expressions of the reference rotor voltages such as:
Urq - A • Ls
- ki • sgn(((Pg J
a Pg-ref + A • Rs
Pg-(ed - ep(-
• Qg +
(eq + ev)-
Urd --A • Ls 22g-ref - A • Rs
- k2 • sgn (s (Qg )
The reference value of the active power exchanged between the wind generator and the grid is generated by MPPT control and it has given by (26). The reference grid reactive power Qg-ref is fixed to zero value to maintain the power factor at unity. The detailed scheme of the studied system is illustrated in Fig. 7.
Fig. 7. The scheme of BDFRG using the sliding mode controller
8. Results and interpretations. The control technique suggested in this paper has been approved by the Matlab/Simulink software. The generator used in this simulation is 4.5 kW. This generator is connected directly to the grid through its primary stator and controlled through its secondary stator. Rated parameters are next [5]: Rp = 3.781 Q, Rs = 2.441 Q, Lp = 0.41 H, Ls = 0.316 H, Lm = 0.3 H, J = 0.2 kg-m2, Pr = 4. Wind turbine parameters are next: blade radius R = 4 m, gearbox ratio G = 7.5, turbine inertia 1.5 kg^m2, air density p = 1.225 kg/m3 and number of blades is 3.
Figure 8 indicates the speed of the wind. The mechanical speed generated by the turbine is similar to the wind profile applied to the turbine. The reference value Pg-ref of the grid active power is determined by (26), and the reference of the reactive power is maintained at zero to guarantee unity power factor.
Fig. 8. Wind speed
Figures 9 demonstrate the performances of the vector control and SMC of the grid active and reactive powers used to a wind turbine mechanism structured from a BDFRG.
e
p
e
e
q
+
e
+
Vector Control Sliding Mode Control
order of harmonic order of harmonic
f f'
Fig. 9. Vector control and sliding mode control of the BDFRG: a and a - mechanical speed; b and b' - electromagnetic torque; c and c' - grid active and reactive power; d and d' - secondary stator phase voltage and current; e and e' - primary stator phase voltage and current; f and f - total harmonic distortion of line current
Both control strategies approve an idealist decoupling between both elements of the BDFRG power (active and reactive). The outcomes found, without any doubt, demonstrate that the usage of the two commands can maintain the active and reactive powers to their aimed values. Figure 9 is the simulation results for active and reactive power response in using MPPT when the
traditional PI controller (Fig. 9,a-e) and sliding mode control (Fig. 9,a'-e') are applied. In this study, simulation results show clearly the improvement of active and reactive power demand obtained by applying sliding mode control in terms of time response and good reference tracking accuracy than those obtained using the traditional PI regulator. In the case of star-up, we notice
that the sliding mode controller transient responses of both active and reactive powers present no overshoot, whereas the steady-state error is close to zero.
Figures 9,d,e and Fig. 9,d',e' present the winding currents in which we observe that both the frequency and the amplitude of these control currents (secondary currents Is) change during the period of variation of active and reactive powers. On the other hand, the frequency of
the current of the supply winding (primary currents Ip) remains constant to be adapted to the supply frequency of the grid, so when the reference of the active power is changed, the amplitude of the current also is changed.
Robustness tests. Figure 10 represents a comparison between the two controllers robustness: PI and SMC with parametric variations Lp, Ls, Rs and Lm of -20 % and +20 % of their nominal values.
Fig. 10. Comparison of robustness between power control with PI and SMC of BDFRG with parametric variations Lp, Ls, Rs and Lm of -20 % (a) and +20 % (b) of their nominal values
From the obtained results (Fig. 10,^ and Fig. 10,6) the SMC strategy is better than the vector control with PI in terms of response time and reference tracking.
Conclusions. This paper has presented a comparative study between two controllers of active and reactive powers for the wind energy system equipped with a brushless doubly fed reluctance generator. The first one is a proportional-integral controller, and the second is a sliding mode controller-based field oriented control strategy.
Simulation results show the optimized performances of the vector control strategy based on a sliding mode controller. We observe high performances in terms of response time for vector control is 0.2 s and for sliding mode is 0.04 s. Spectral analysis of line current shows that total harmonic distortion of vector control is 27.25 % unlike sliding mode able to reduce the total harmonic distortion to a low value of around 16.25 % (without filter) and reference tracking without overshoots through the response characteristics. The decoupling, the stability, and the convergence towards the equilibrium are assured. Furthermore, this regulation presents a high dynamic
response, and it is more robust against parameter variation of the brushless doubly fed reluctance generator versus the conventional proportional-integral controller.
Conflict of interest. The authors declare that they have no conflicts of interest.
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Received 25.10.2021 Accepted 30.11.2021 Published 23.02.2022
Oualah Oussama1, PhD Student, Djallel Kerdoun1, Professor, Abderraouf Boumassata1, Dr.,
1 Electrical Engineering Laboratory of Constantine (LGEC), Department of Electrical Engineering,
Mentouri Brothers University, Constantine 1, Campus Ahmed Hamani Zerzara, Constantine, 25000, Algeria
e-mail: [email protected] (Corresponding author); [email protected]
2 Department of Electronics, Electrical Engineering and Automation,
Laboratory of Electrical Engineering of Polytechnic School of Constantine (LGEPC),
National Polytechnic School of Constantine, Algeria e-mail: [email protected]
How to cite this article:
Qualah O., Kerdoun D., Boumassata A. Comparative study between sliding mode control and the vector control of a brushless doubly fed reluctance generator based on wind energy conversion systems. Electrical Engineering & Electromechanics, 2022, no. 1, pp. 51-58. doi: https://doi.org/10.20998/2074-272X.2022.L07.
